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<div id="Sx1" class="ltx_section">
<h1 class="ltx_title ltx_title_section">Localization uncertainty</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">Let <img id="Sx1.p1.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi105.png" width="15" height="12" alt="\sigma"> be the standard deviation of a fitted Gaussian PSF in
nm, <img id="Sx1.p1.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi107.png" width="13" height="12" alt="a"> is the (backprojected) pixel size in nm, <img id="Sx1.p1.m3" class="ltx_Math" style="vertical-align:-2px" src="mi/mi18.png" width="20" height="16" alt="N"> is the number
of photons detected for a given molecule, and <img id="Sx1.p1.m4" class="ltx_Math" style="vertical-align:-2px" src="mi/mi108.png" width="12" height="16" alt="b"> is the background
signal level in photons calculated as the standard deviation of the
residuals between the raw data and the fitted PSF model <cite class="ltx_cite">[<a href="#bib.bib18" title="Minimizing detection errors in single molecule localization microscopy" class="ltx_ref">1</a>]</cite>.
The uncertainty in the lateral position of a molecule can be approximated
by the formula <cite class="ltx_cite">[<a href="#bib.bib38" title="Precise nanometer localization analysis for individual fluorescent probes" class="ltx_ref">3</a>]</cite></p>
<table id="Sx1.E1" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E1.m1" class="ltx_Math" style="vertical-align:-14px" src="mi/mi104.png" width="265" height="42" alt="\left\langle(\Delta x)^{2}\right\rangle=\frac{\sigma^{2}+a^{2}/12}{N}+\frac{8%
\pi\sigma^{4}b^{2}}{a^{2}N^{2}}\,."></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(1)</span></td>
</tr>
</table>
</div>
<div id="Sx1.p2" class="ltx_para">
<p class="ltx_p">The previous equation can be further adjusted to take EM gain of EMCCD
cameras into account <cite class="ltx_cite">[<a href="#bib.bib27" title="Localization capability and limitation of electron-multiplying charge-coupled, scientific complementary metal-oxide semiconductor, and charge-coupled devices for superresolution imaging" class="ltx_ref">2</a>]</cite>, giving the expression</p>
<table id="Sx1.E2" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.E2.m1" class="ltx_Math" style="vertical-align:-14px" src="mi/mi103.png" width="274" height="42" alt="\left\langle(\Delta x)^{2}\right\rangle=\frac{2\sigma^{2}+a^{2}/12}{N}+\frac{8%
\pi\sigma^{4}b^{2}}{a^{2}N^{2}}\,."></td>
<td class="ltx_eqn_pad"></td>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(2)</span></td>
</tr>
</table>
</div>
<div id="Sx1.p3" class="ltx_para">
<p class="ltx_p">In the 3D case, the localization uncertainty of molecular positions
in the lateral direction is determined according to Equations (<a href="#Sx1.E2" title="(2) ‣ Localization uncertainty" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>)
or (<a href="#Sx1.E1" title="(1) ‣ Localization uncertainty" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>), where we use <img id="Sx1.p3.m1" class="ltx_Math" style="vertical-align:-6px" src="mi/mi106.png" width="151" height="23" alt="\sigma^{2}=\sigma_{1}\left(\theta_{z}\right)\sigma_{2}\left(\theta_{z}\right)">.
In the axial direction, we use a constant, user-specified value.</p>
</div>
<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<ul id="I1" class="ltx_itemize">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="PSF.html" title="" class="ltx_ref">Point-spread function (PSF)</a></p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="Fitting.html" title="" class="ltx_ref">Fitting point-spread function models</a></p>
</div>
</li>
<li id="I1.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i3.p1" class="ltx_para">
<p class="ltx_p"><a href="../../calibration/DefocusModels.html" title="" class="ltx_ref">Defocusing models</a></p>
</div>
</li>
<li id="I1.i4" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i4.p1" class="ltx_para">
<p class="ltx_p"><a href="../../CameraSetupPlugIn.html" title="" class="ltx_ref">Camera setup</a></p>
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<div id="bib" class="ltx_bibliography">
<h1 class="ltx_title ltx_title_bibliography">References</h1>

<ul id="L1" class="ltx_biblist">
<li id="bib.bib18" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[1]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">P. Křížek, I. Raška and G. M. Hagen</span><span class="ltx_text ltx_bib_year">(2011)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Minimizing detection errors in single molecule localization microscopy</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Optics Express</span> <span class="ltx_text ltx_bib_volume">19</span> (<span class="ltx_text ltx_bib_number">4</span>), <span class="ltx_text ltx_bib_pages"> pp. 3226–35</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1364/OE.19.003226" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="Localization uncertainty" class="ltx_ref"><span class="ltx_text ltx_ref_title">Localization uncertainty</span></a>.
</span>
</li>
<li id="bib.bib27" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[2]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">T. Quan, S. Zeng and Z.-L. Huang</span><span class="ltx_text ltx_bib_year">(2010)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Localization capability and limitation of electron-multiplying charge-coupled, scientific complementary metal-oxide semiconductor, and charge-coupled devices for superresolution imaging</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Journal of Biomedical Optics</span> <span class="ltx_text ltx_bib_volume">15</span> (<span class="ltx_text ltx_bib_number">6</span>), <span class="ltx_text ltx_bib_pages"> pp. 066005</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1117/1.3505017" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p2" title="Localization uncertainty" class="ltx_ref"><span class="ltx_text ltx_ref_title">Localization uncertainty</span></a>.
</span>
</li>
<li id="bib.bib38" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[3]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">R. E. Thompson, D. R. Larson and W. W. Webb</span><span class="ltx_text ltx_bib_year">(2002)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Precise nanometer localization analysis for individual fluorescent probes</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Biophysical Journal</span> <span class="ltx_text ltx_bib_volume">82</span> (<span class="ltx_text ltx_bib_number">5</span>), <span class="ltx_text ltx_bib_pages"> pp. 2775–83</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1016/S0006-3495(02)75618-X" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="Localization uncertainty" class="ltx_ref"><span class="ltx_text ltx_ref_title">Localization uncertainty</span></a>.
</span>
</li>
</ul>
</div>
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